President Donald Trump speaks during a press conference to discuss a revised U.S. trade agreement with Mexico and Canada in the Rose Garden of the White House on October 1, 2018 in Washington, DC.

There is still plenty of confusion about the FBI's renewed background investigation of embattled Supreme Court nominee Brett Kavanaugh, even after President Donald Trump was asked to clear it up Monday.

The president said that "it wouldn't bother me at all" if the FBI interviewed all three women who have publicly accused Trump's nominee, Kavanaugh, 53, of sexual misconduct as a young man.

Yet Trump, who spoke during a news conference in the White House's Rose Garden, also reiterated to reporters that he believes that the Senate should decide how broad the FBI's probe of the accusations against Kavanaugh should be.

"I want them to do a very comprehensive investigation, whatever that means according to the senators and the Republicans and the Republican majority," said the president, who ultimately controls how expansive the probe can be.

"My White House is doing whatever the senators want," he added.

But even as he said that, it was still unclear whether White House counsel Don McGahn will or has directed the FBI to expand its ongoing inquiry into all three women's claims, and into allegations that Kavanaugh lied about the extent and effects of his admitted drinking in high school and college.

The confusion was heightened by the fact that Trump's comments about the Kavanaugh case and its scope came in a rambling series of remarks in which he both suggested there should be no limitation placed on the investigation by the White House, but also implied that agents might not question all of the self-identified accusers.

However, The New York Times reported later Monday, citing two people briefed on the issue, that the White House had authorized the FBI to expand its probe "by interviewing anyone it deems necessary as long the review is finished by the end of the week."

NBC News has confirmed that report.

Limited witness list

Also Monday, nine Democratic senators on the Senate Judiciary Committee sent McGahn and FBI Director Christopher Wray a letter listing 24 people and entities they believe the FBI should speak to as part of the probe.

The letter noted that it was being sent in light of reports that the White House is limiting the scope of the investigation into sexual assault allegations against" Kavanaugh.

That is 20 more people than the White House had reportedly told the FBI to question.

As of Monday, three days after the FBI was told to reopen its background probe, agents have not interviewed Christine Blasey Ford or Julie Swetnick, two of Kavanaugh's accusers, and the ones whose claims against him are the most serious.

Neither Ford nor Swetnick was on the original list of people to be questioned.

Kavanaugh, who has strongly denied all claims of sexual misconduct, himself is not on the list of people to be questioned by the FBI in its probe.

The probe was set in motion after Sen. Jeff Flake, R-Ariz., agreed to vote with other GOP senators on the Judiciary Committee to advance Kavanaugh's nomination to the full Senate, with the provision that he wanted the FBI to investigate newly raised allegations against him.

Flake, in an interview aired Sunday night, told the CBS program "60 Minutes" that if the FBI finds that Kavanaugh lied to the committee during his confirmation hearings, his nomination will be doomed.

On Monday, Flake said, "We certainly want the FBI to do a real investigation, and are working to make sure that happens." He said he had been talking to the White House counsel's office about the probe's scope to ensure "that any current, credible allegation is fully investigated."

"It does us no good to do an investigation that just gives us more cover," Flake said.

Trump on Saturday night had said on Twitter that the FBI should question whoever agents consider necessary for their investigation.

But The New York Times reported Saturday that the White House had listed just four people on the witness list for the FBI to question: Kavanaught's high school drinking buddy Mark Judge, Leland Keyser, P.J. Smyth and Deborah Ramirez.

NBC News reported Sunday that the FBI had received no new instructions about the scope of its probe after Trump's tweet.

The accusers

Ford has testified to the Judiciary Committee that Judge was in a room with her and Kavanaugh during the alleged assault at an early 1980s gathering in a house in suburban Maryland. She claims that Kavanaugh allegedly held her down on a bed, covered her mouth as she tried to scream, and grinded his body against her as he tried to rip off her clothes. Ford also says Keyser and Smyth were elsewhere in the house.

Ramirez, likewise, told The New Yorker that Kavanaugh exposed himself to her during a drunken dorm party at Yale in the mid-1980s.

Swetnick, for her part, said she attended house parties in the early 1980s where she saw Kavanaugh and Judge spike the drinks being served to lower the inhibitions of girls there so it would be easier for them to be gang raped.

The list did not include a growing number of people who have come forward to cast into doubt Kavanaugh's testimony that he never would have blacked out, or forgotten what happened, during periods of heavy drinking.

Trump told reporters on Monday, "I think the FBI should interview anybody they want, within reason. But you have to say within reason."

"I don't know all three of the accusers. Certainly imagine they're going to interview two. The third one, I don't know much about," Trump said, without naming the woman.

"But it wouldn't bother me at all. I mean. I've heard that the third one has — I have no idea if this is true — very little credibility. If there is any credibility, interview the third one," Trump said.

The proposed additional witnesses sent by the Democratic senators to the White House on Monday include Ford, Swetnick and Kavanaugh, as well as friends of Ford, Kavanaugh and Judge.